Number System Divisibility by 7

Question

X is a number formed by writing 9 for 99 times. What will be the remainder of this number when divided by 7?

Answer 1

$X= 10^{99} - 1 = 10^{3 \times 33} - 1$.

Now $10^3 = 1000 = -1$ modulo $7$.

So $X=(-1)^{33} - 1 = - 1 - 1 = 5$ modulo $7$.